An enhanced nonlinear PID controller design for the CSTR system utilizing NARX-Laguerre model

Abstract

This study leverages a second-order discrete-time Nonlinear AutoRegressive Exogenous (NARX)-Laguerre model to develop a nonlinear predictive proportional–integral–derivative (PID) control approach. The proposed technique employs the model predictive control (MPC) framework to dynamically adjust the PID controller parameters. This work is motivated by the limitations of conventional PID controllers in handling nonlinear dynamics and system constraints, which are frequently encountered in real-world applications. In addition, the challenge of control signal saturation is addressed, with a practical solution introduced to improve system performance. The effectiveness of this advanced PID control strategy is demonstrated through its application to a continuous stirred tank reactor (CSTR).

Keywords

PID optimization model predictive control simulation nonlinear control

Introduction

Advancements in system control theory have yielded a myriad of controllers and methods for application in real systems. Among these, proportional–integral–derivative (PID) controllers have stood out for their simplicity and effectiveness across a wide range of industrial systems (Huang et al., 2004; Ikuro et al., 2013; O’Dwyer, 2009; So, 2024). The popularity of PID controllers arises from their ability to provide satisfactory performance for systems with reduced-order structures.

However, real-world systems often exhibit nonlinear behavior and are subject to operational constraints, which can significantly degrade the performance of conventional PID controllers. In such cases, fixed-parameter PID control is insufficient to ensure robust performance and constraint handling. These limitations motivate the integration of predictive strategies into the PID framework to enhance its adaptability and efficiency in complex scenarios.

The optimization of a PID controller involves fine-tuning its parameters, a process crucial for achieving desired control outcomes. Various analytical and numerical methods have been employed for PID parameter setting (Meric and Serdar, 2015; Na, 2001; Zhao et al., 2023). While these controllers are commonly embraced for their simplicity and effectiveness, their deficiency in predictive capabilities restricts their applicability to intricate systems characterized by complex dynamics. Consequently, there is a drive to incorporate predictive elements into the PID control structure (Na, 2001; Wang et al., 2017).

In this context, our methodology is influenced by model predictive control (MPC), wherein the adjustment of PID controller parameters is conceptualized as a minimization problem. The goal is to focus on minimizing the squared deviation between the reference signal and the predicted output within a specified prediction horizon. MPC is a well-established real-time control strategy that iteratively computes the optimal control actions by solving an optimization problem over a predefined future horizon, while accounting for specific process constraints (Dewei and Yugeng, 2011; Mbarek et al., 2015; Michael et al., 2015; Salhy et al., 2013). The strategy can be adapted to diverse system configurations through explicitly using a model to formulate control laws. The literature presents various MPC extensions, including multi-input-multi-output (MIMO) MPC (Abdelwahed et al., 2017; Mbarek et al., 2015; Wibowo and Saad, 2017), nonlinear model predictive control (NMPC) (Grancharova and Johansen, 2012; Nikolakopolou and Braatz, 2023; Xiang et al., 2006), and MIMO nonlinear MPC (Lopez-Sanz et al., 2017).

Within the context of NMPC, the NARX model (Nonlinear AutoRegressive Exogenous model) is a common choice for constructing nonlinear controllers, as noted in (Napoli and Piroddi, 2010; Nikolakopolou and Braatz, 2023; Xiang et al., 2006). The enhanced precision and increased availability of experimental data in engineering have spurred the development of detailed mathematical models for deeper insights into system behavior. However, representing nonlinear processes with the NARX model poses difficulties, primarily due to its high parameter count, which not only complicates the NMPC algorithm but may also exceed the allowable computation time within the sampling periods of some nonlinear systems.

Recent advancements suggest a new approach to tackle the limitations in representing complex dynamic linear systems. This involves utilizing Laguerre bases to reduce the parameter complexity of various models, including ARX (Njima and Garna, 2020), bilinear models (Dai et al., 2021; Garna et al., 2014), MIMO ARX (Mbarek et al., 2015), and multiple models (Hajer et al., 2020; Sameh et al., 2018). The method focuses on projecting model parameters onto independent Laguerre orthonormal bases to streamline the parameter set. In the nonlinear context, Benabdelwahed et al. propose reducing the NARX model’s complexity by developing it on five distinct Laguerre orthonormal bases (Benabdelwahed et al., 2018). This model, called the NARX-Laguerre model, incorporates five poles, and their optimal configuration plays a key role in significantly decreasing the number of parameters.

Moreover, the integration of the NARX-Laguerre model within the MPC framework provides distinct advantages. Compared to the traditional NARX model, the NARX-Laguerre model reduces the parameter complexity by projecting nonlinearities onto Laguerre bases, resulting in a more compact and computationally efficient parameter set. This simplification significantly reduces the computational burden and enhances real-time feasibility, especially in systems with fast dynamics or limited computational resources. In addition, the NARX-Laguerre model retains the ability to accurately capture nonlinear system behaviors, which is critical for effective PID control in complex systems, offering both stability and predictive power.

While MPC brings a wealth of predictive advantages, the marriage of its predictive strengths with the stability and robustness of PID control creates a formidable synergy. The advantages of PID simplicity, stability, and effectiveness are strategically combined with the predictive prowess of MPC. This hybridization retains the reliability of PID while imbuing it with the foresight to preemptively address potential deviations from optimal system performance.

The main contribution presented in this paper is the synthesis of a predictive control strategy for nonlinear systems, specifically building upon the foundations of the PID controller. The predictive element is derived from the implementation of NMPC, formulated using NARX-Laguerre model. In this approach, the adjustment of PID parameters is integrated into the optimization process for determining the optimal control sequence. The optimization of the Laguerre poles, critical parameters of this model, is achieved through the application of the Genetic Algorithm (GA). To validate the effectiveness of the proposed control methodology, comprehensive testing is conducted using the continuous stirred tank reactor (CSTR) benchmark. The paper is structured as follows: section “Second-order NARX-Laguerre model” introduces the NARX-Laguerre model, a simplified version of the NARX model with reduced parameterization, presented in a recursive format. A GA is employed to fine-tune the five Laguerre poles. Section “PID controller based on NARX-Laguerre model” describes the development of a predictor that uses both future control actions and predicted outputs over the predictive horizon. The NMPC approach utilizing the NARX-Laguerre model and incorporating system constraints, is then synthesized. Finally, the performance of the proposed NARX-Laguerre model and its NMPC application is demonstrated using a CSTR benchmark case study.

Second-order NARX-Laguerre model

Definition

Introduced by Benabdelwahed et al. (2018), the NARX-Laguerre model is derived by projecting the coefficients of NARX model onto a set of five distinct Laguerre bases $ℑ_{u} = {ℓ_{n}^{u} (ξ_{u})}_{n = 0}^{\infty}$ , $ℑ_{y} = {ℓ_{n}^{y} (ξ_{y})}_{n = 0}^{\infty}$ , $ℑ_{uu} = {ℓ_{n}^{uu} (ξ_{uu})}_{n = 0}^{\infty}$ , $ℑ_{yy} = {ℓ_{n}^{yy} (ξ_{yy})}_{n = 0}^{\infty}$ , and $ℑ_{uy} = {ℓ_{n}^{uy} (ξ_{uy})}_{n = 0}^{\infty}$ .

This model is marked by fewer parameters compared to the classical model and defined by the following expression:

\begin{matrix} y (k) = \sum_{n = 0}^{N_{u} - 1} g_{n}^{u} x_{n}^{u} (k) + \sum_{n = 0}^{N_{y} - 1} g_{n}^{y} x_{n}^{y} (k) + \sum_{n_{1} = 0}^{N_{u} - 1} \sum_{n_{2} = 0}^{N_{u} - 1} g_{n_{1} n_{2}}^{uu} . s_{n_{1} n_{2}}^{uu} (k) \\ + \sum_{n_{1} = 0}^{N_{y} - 1} \sum_{n_{2} = 0}^{N_{y} - 1} g_{n_{1} n_{2}}^{yy} s_{n_{1} n_{2}}^{yy} (k) + \sum_{n_{1} = 0}^{N_{u} - 1} \sum_{n_{2} = 0}^{N_{y} - 1} g_{n_{1} n_{2}}^{uy} x_{n_{1} u}^{uy} (k) x_{n_{2} y}^{uy} (k) \end{matrix}

(1)

where $y (k)$ and $u (k)$ are the output and input signals, respectively, $(N_{y} \geq 1)$ and $(N_{u} \geq 1)$ represent the truncation orders related to the output and input, respectively, and ( $g_{n}^{u}$ , $g_{n}^{y}$ , $g_{n_{1} n_{2}}^{uu}$ , $g_{n_{1} n_{2}}^{yy}$ and $g_{n_{1} n_{2}}^{uy}$ ) are the Fourier coefficients. The output filters $x_{n}^{u} (k)$ , $x_{n}^{y} (k)$ , $x_{n_{1 u}}^{uy} (k)$ , and $x_{n_{2 y}}^{uy} (k)$ and the two expressions $s_{n_{1} n_{2}}^{uu}$ and $s_{n_{1} n_{2}}^{yy}$ are expressed using Laguerre functions $ℓ_{i}^{n}$ and the parameters $ξ_{i}, (i = u, y, uu, yy, uy)$ characterizing Laguerre bases, called the “Laguerre poles,” as detailed in Benabdelwahed et al. (2018). The NARX-Laguerre model offers two major advantages that make it highly suitable for nonlinear system modeling and control. First, by projecting the input and output regressors onto orthonormal Laguerre bases, it significantly reduces the number of parameters required to accurately capture system dynamics. This reduction not only simplifies model identification but also improves computational efficiency—an essential aspect when embedded in real-time control applications. Second, the NARX-Laguerre structure retains the nonlinear approximation capabilities of standard NARX models while offering better generalization and robustness against overfitting due to its compact parameterization.

Recursive representation of the NARX-Laguerre model

Recursively, the filtered input and output of the NARX-Laguerre model can be expressed through the following relations:

{\begin{matrix} X_{N_{u}}^{u} (k + 1) = A_{u} X_{N_{u}}^{u} (k) + b_{u} u (k) \\ X_{N_{y}}^{y} (k + 1) = A_{y} X_{N_{y}}^{y} (k) + b_{y} y (k) \\ X_{u, N_{u}}^{uy} (k + 1) = A_{uy, u} X_{u, N_{u}}^{uy} (k) + b_{uy, u} u (k) \\ X_{y, N_{y}}^{uy} (k + 1) = A_{uy, y} X_{y, N_{y}}^{uy} (k) + b_{uy, y} y (k) \\ X_{N_{u}}^{uu} (k + 1) = A_{uu, u} X_{N_{u}}^{uu} (k) + b_{uu, u} u (k) \\ X_{N_{y}}^{yy} (k + 1) = A_{yy, y} X_{N_{y}}^{yy} (k) + b_{yy, y} y (k) \end{matrix}

(2)

where the matrices $A_{t, p}$ and the vectors $b_{t, p}$ ( $(t, p) = (u, u)$ , $(y, y)$ , $(uu, u)$ , $(yy, y)$ , $(uy, u)$ , $(uy, y)$ ) are defined in Ben Abdelwahed et al. (2023) and Benabdelwahed et al. (2018).

Then, the following recursive expressions are formulated:

{\begin{matrix} S_{uu, N_{u}} (k + 1) = A_{s, u} S_{uu, N_{u}} (k) + b_{uu, u} \otimes X_{N_{u}}^{uu} (k) u (k) \\ S_{yy, N_{y}} (k + 1) = A_{s, y} S_{yy, N_{y}} (k) + b_{yy, y} \otimes X_{N_{y}}^{yy} (k) y (k) \end{matrix}

(3)

where

\begin{matrix} X_{N_{u}}^{uu} (k) & = [x_{0}^{uu} (k) x_{1}^{uu} (k) \dots x_{N_{u} - 1}^{uu} (k)] \end{matrix}

(4)

\begin{matrix} X_{N_{y}}^{yy} (k) & = [x_{0}^{yy} (k) x_{1}^{yy} (k) \dots x_{N_{y} - 1}^{yy} (k)] \end{matrix}

(5)

and ⊗ is the Kronecker product. The NARX-Laguerre model satisfies the following recursive vector representation:

{\begin{matrix} {\begin{matrix} X_{N_{u}}^{u} (k + 1) = A_{u} X_{N_{u}}^{u} (k) + b_{u, u} u (k) \\ X_{N_{y}}^{y} (k + 1) = A_{y} X_{N_{y}}^{y} (k) + b_{y, y} y (k) \\ S_{uu, N_{u}} (k + 1) = {\underset{̲}{A}}_{u} S_{uu, N_{u}} (k) + b_{uu, u} \otimes X_{N_{u}}^{uu} (k) u (k) \\ S_{yy, N_{y}} (k + 1) = {\underset{̲}{A}}_{y} S_{yy, N_{y}} (k) + b_{yy, y} \otimes X_{N_{y}}^{yy} (k) y (k) \\ X_{u, N_{u}}^{uy} (k + 1) = A_{uy, u} X_{u, N_{u}}^{uy} (k) + b_{uy, u} u (k) \\ X_{y, N_{y}}^{uy} (k + 1) = A_{uy, y} X_{y, N_{y}}^{uy} (k) + b_{uy, y} y (k) \end{matrix} \\ y (k) = C_{u}^{T} X_{N_{u}}^{u} (k) + C_{y}^{T} X_{N_{y}}^{y} (k) + C_{uu}^{T} S_{uu, N_{u}} (k) + \\ C_{yy}^{T} S_{yy, N_{y}} (k) + C_{uy}^{T} P^{uy} (k) \end{matrix}

(6)

NARX-Laguerre model can be written in the following linear-in-parameter form:

\begin{matrix} y (k) & = C^{T} V \end{matrix}

(7)

\begin{matrix} C & = {[C_{u}^{T} C_{y}^{T} C_{uu}^{T} C_{yy}^{T} C_{uy}^{T}]}^{T} \end{matrix}

(8)

where $C_{u}$ , $C_{y}$ , $C_{uu}$ , $C_{yy}$ , and $C_{uy}$ are the parameter vectors containing the Fourier coefficients, defined by:

{\begin{matrix} C_{u} = {[g_{0}^{u}, g_{1}^{u}, . . g_{N_{u} - 1}^{u}]}^{T} \\ C_{y} = {[g_{0}^{y}, g_{1}^{y}, . . g_{N_{y} - 1}^{y}]}^{T} \\ C_{uu} = {[g_{0, 0}^{uu}, g_{0, 1}^{uu}, . ., g_{0, N_{u} - 1}^{uu}, g_{1, 0}^{uu}, . ., g_{N_{u} - 1, N_{u} - 1}^{uu}]}^{T} \\ C_{yy} = {[g_{0, 0}^{yy}, g_{0, 1}^{yy}, . ., g_{0, N_{y} - 1}^{yy}, g_{1, 0}^{yy}, . ., g_{N_{y} - 1, N_{y} - 1}^{yy}]}^{T} \\ C_{uy} = {[g_{0, 0}^{uy}, g_{0, 1}^{uy}, . ., g_{0, N_{u} - 1}^{uy}, g_{1, 0}^{uy}, . ., g_{N_{u} - 1, N_{y} - 1}^{uy}]}^{T} \end{matrix}

(9)

and

V = {[X_{N_{u}}^{u} (k), X_{N_{y}}^{y} (k), S_{uu, N_{u}} (k), S_{yy, N_{y}} (k), P^{uy} (k)]}^{T}

(10)

with $S_{uu, N_{u}} (k)$ , $S_{yy, N_{y}} (k)$ , $X_{N_{u}}^{u} (k)$ , $X_{N_{y}}^{y} (k)$ , and $P^{uy} (k)$ are detailed in Benabdelwahed et al. (2018).

Pole optimization using GA

Due to its straightforwardness and efficiency, the GA is commonly employed as an optimization technique. This approach revolves around minimizing or maximizing a criterion known as “fitness” to attain the desired optimal variables. To identify the optimum values of the Laguerre poles, we opt for the application of GAs, focusing on minimizing the normalized mean square error (NMSE) defined by:

NMSE = \frac{\sum_{k = 1}^{M} {[y_{m} (k) - y (k)]}^{2}}{\sum_{k = 1}^{M} {[y_{m} (k)]}^{2}}

(11)

where $y_{m}$ and y are, respectively, the measured and the model outputs.

* Initialization: During this phase, an “initial population” of elements that will be optimized is formed using randomly generated values.

* Rating: The likelihood of minimizing the “fitness” function assigned to each element or group of elements is calculated.

* Selection: Elements from the population are selected based on their computed probabilities in the preceding phase.

* Cross over: Using two elements randomly chosen from the population named “parents,” the genetic recombination yields two new elements referred to as “children.”

* Mutation: The “children” experience a modification in this stage. The purpose of the mutation is to avert the algorithm from converging to a local extremum.

PID controller based on NARX-Laguerre model

Methodology

To enhance the control law, the NMPC approach is utilized, accounting for the system’s future behavior as predicted by an accurate mathematical model. This technique is incorporated into the design of a PID controller. The control scheme is depicted in Figure 1.

Figure 1.

Block diagram of PID NMPC controller.

A PID controller computes the control law $u (t)$ based on the error signal $e (t)$ , which quantifies the deviation of the system output $y (t)$ from the reference signal $r (t)$ , using $e (t) = r (t) - y (t)$ :

\begin{matrix} u_{PID} (t) = k_{P} (e (t) + k_{I} \int_{0}^{t} e (τ) dτ + k_{D} ė (t)) \end{matrix}

(12)

where $k_{P}$ is the proportional gain that controls the immediate reaction to the error, influencing how quickly the system corrects itself, $k_{I}$ is the integral gain addresses past accumulated errors, ensuring the system eliminates steady-state error, although it can introduce slower responses and $k_{D}$ is the derivative gain that predicts future errors by analyzing the rate of change, enhancing stability and minimizing overshoot. Proper tuning of these three gains is essential for achieving a responsive, accurate, and stable system performance. However, the fixed gains of PID controllers lead to sensitivity to disturbances and ineffectiveness in handling parameter uncertainties or time-varying dynamics.

Cost function

The method used for control calculation relies on the quadratic norm, as presented in Maraoui et al. (2019). The primary goal is to minimize the squared error between the predicted outputs ŷ and the reference setpoints r. It is assumed that the reference values are available for both the current time step k and throughout the prediction horizon. The quadratic criterion is formulated in equation (13):

\begin{matrix} J (k) & = μ_{1} \sum_{i = 0}^{N_{p} - 1} ({(y (k + i) - r (k + i))}^{2} + μ_{2} \sum_{i = 0}^{N_{p} - 2} u {(k + i)}^{2} \\ + μ_{3} [{(k_{P} (k) - k_{P} (k - 1))}^{2} + {(k_{I} (k) - k_{I} (k - 1))}^{2} \\ + {(k_{D} (k) - k_{D} (k - 1))}^{2}] \end{matrix}

(13)

where $N_{p}$ represents the prediction horizon, and the weighting coefficients $μ_{1}$ , $μ_{2}$ , and $μ_{3}$ are defined as follows:

$μ_{1}$ penalizes the tracking error between the reference and predicted outputs,

$μ_{2}$ penalizes the control signal magnitude,

$μ_{3}$ penalizes the increments of the PID parameters.

The cost function, $J (k)$ , comprises four components: The initial term accounts for the sum of squared errors between the predicted output and the desired output. The second term involves the energy supplied by the control signals. The third term is incorporated to stabilize the PID controller parameters.

j-step ahead predictor

According to equation (7), $y (k)$ can be expressed in an incremental way as follows:

\begin{matrix} \begin{matrix} y (k) & = y (k - 1) + δy (k) \\ = y (k - 1) + C^{T} δϑ (k) \end{matrix} \end{matrix}

(14)

where $δϑ (k)$ is defined as:

δϑ (k) = ϑ (k) - ϑ (k - 1)

(15)

The j-step ahead predictor of the NARX-Laguerre model, denoted as $ŷ (k + j ∕ k)$ , obtained by iteratively substituting $y (k)$ is expressed as follows:

y (k + j ∕ k) = y (k) + C^{T} \sum_{i = 1}^{j} δϑ (k + i)

(16)

Such that $δϑ (k + j)$ represents the j-step ahead predictor of $δϑ (k)$ , derived from relation (10), and is given by the following expression:

\begin{matrix} \begin{matrix} δϑ (k + j) & = [{(δ X^{u} (k + j))}^{T}, {(δ X^{y} (k + j))}^{T}, {(δ S_{uu} (k + j))}^{T}, \\ {{(δ S_{yy} (k + j))}^{T}, {(δP (k + j))}^{T}]}^{T} \end{matrix} \end{matrix}

(17)

The j-step ahead predictor is detailed in Ben Abdelwahed and Bouzrara (2024).

Matrix form

In matrix form, the criterion (13) can be written as follows:

\begin{matrix} \begin{matrix} J (K (k)) & = μ_{1} {(Y_{pred} (k) - R (k))}^{T} (Y_{pred} (k) - R (k)) + μ_{2} U (k) U {(k)}^{T} \\ + μ_{3} {(K (k) - K (k - 1))}^{T} (K (k) - K (k - 1)) \end{matrix} \end{matrix}

(18)

where $Y_{pred} (k)$ is an $(N_{p})$ -dimensional vector of predicted outputs and $R (k)$ is the vector of reference trajectory within the prediction horizon $[k, k + N_{p} - 1]$ defined respectively as follows:

\begin{matrix} Y_{pred} (k) & = {[y (k), \dots, y (k + N_{p} - 1)]}^{T}, \\ R (k) & = {[r (k), \dots, r (k + N_{p} - 1)]}^{T} \end{matrix}

(19)

$U (k)$ is the control sequence within the control horizon $[k, k + N_{p} - 2]$ defined as follows:

U (k) = {[u (k), \dots, u (k + N_{p} - 2)]}^{T}

(20)

K is defined as follows:

K (k) = {[K_{P} (k) K_{D} (k) K_{I} (k)]}^{T}

(21)

The desired control signal $u_{PID} (k)$ given by equation (12) can be written in matrix form as follows:

u (k) = E (k) K (k) + K_{Ref} (k) E (k)

(22)

where the vectors $E (k)$ and $K_{Ref} (k)$ are defined as follows:

E (k) = [e (k) se (k) de (k)]

(23)

such that $se (k) = \sum_{h = 1}^{k} e (h)$ and $de (k) = (e (k) - e (k - 1)$ .

Constraints

Minimizing $J (K (k))$ determines the future values of K which are essential for computing the optimal future control $u (k)$ . This step depends on the predicted output of the model. This latter is then calculated within the prediction horizon $N_{p}$ , by defining a generalized form of the predictor. As previously stated, physical systems operate in practice within various constraints, and therefore, the optimization problem needs to incorporate these limitations. The following inequality presents the input constraints:

\begin{matrix} u_{m} \leq u (k) \leq u_{M} \end{matrix}

(24)

Based on relation (23), the constraints on $u (k)$ given by equation (24) can be written as follows:

\begin{matrix} u_{m} \leq E (k) K (k) + K_{Ref} (k) E (k) \leq u_{M} \end{matrix}

(25)

which gives:

\begin{matrix} E (k) K (k) \leq u_{M} - K_{Ref} E (k) \end{matrix}

(26)

and

\begin{matrix} - E (k) K (k) \leq K_{Ref} E (k) - u_{m} \end{matrix}

(27)

Hence, the constraints on the commands and the increments of commands can be written in the following matrix form:

Γ K (k) \leq V

(28)

with:

Γ = (\begin{matrix} E (k) \\ - E (k) \end{matrix}), V = (\begin{matrix} u_{M} - K_{Ref} E (k) \\ K_{Ref} E (k) - u_{m} \end{matrix})

(29)

Thus, the future control $u (k)$ is derived by solving the following optimization problem subject to constraints:

\begin{matrix} \min_{K (k) \in} (J (K (k))) \end{matrix}

(30)

\begin{matrix} Ψ (K (k)) = {K (k) ∕ Γ K (k) \leq V} \end{matrix}

(31)

Application of the enhanced PID controller to the benchmark CSTR

Description of the benchmark

The CSTR is a chemical reactor that uses an irreversible and exothermic process to transform a reactant A into a product B (Assala et al., 1997). The process, illustrated in Figure 2, is cooled by a single coolant stream.

Figure 2.

CSTR process.

The two handled inputs of the benchmark are as follows:

The process flow rate q.

The coolant flow rate $q_{c}$ .

The two outputs are as follows:

The effluent concentration $C_{A}$ .

The temperature T.

The CSTR is described by the following equations (32) and (33), with parameters outlined in Table 1:

\begin{matrix} \frac{d C_{A}}{dt} & = \frac{q}{V} (C_{Af} - C_{A}) - k_{0} C_{A} \exp (- \frac{E}{RT}) ϕ_{c} (t) \end{matrix}

(32)

\begin{matrix} \frac{dT}{dt} & = \frac{q}{V} (T_{f} - T) + [\frac{(- Δ H) k_{0} C_{A}}{ρ C_{p}}] \exp (- \frac{E}{RT}) ϕ_{c} (t) \\ + \frac{ρ_{c} C_{pc}}{ρ C_{p} V} q_{c} [1 - \exp (- \frac{hA}{q_{c} ρ C_{pc}} ϕ_{h} (t))] \times (T_{cf} - T) \end{matrix}

(33)

Table 1.

Steady-state operating data.

Parameters	Symbols	Values
Feed flow rate	q	$100 lmi n^{- 1}$
Feed concentration	$C_{Af}$	$1 mol l^{- 1}$
Feed temperature	$T_{f}$	$350 K$
Cooling inlet temperature	$T_{cf}$	$350 K$
CSTR volume	V	$100 l$
Heat transfer term	hA	$7 \times 1 0^{5} calmi n^{- 1} K^{- 1}$
Reaction rate constant	$k_{0}$	$7.2 \times 1 0^{10} mi n^{- 1}$
Activation energy term	$\frac{E}{R}$	$9.95 \times 1 0^{3}$
Heat of reaction	$- Δ H$	$2 \times 1 0^{5} calmo l^{- 1}$
Liquid densities	$ρ, ρ_{c}$	$1000 g l^{- 1}$
Specific heat	$C_{p}, C_{pc}$	$1 cal g^{- 1} K^{- 1}$
Coolant flow rate	$q_{c}$	$103.41 lmi n^{- 1}$
Reactor temperature	T	$440.2 K$
Effluent concentration	$C_{A}$	$8.36 \times 1 0^{- 2} mol l^{- 1}$
Fouling coefficient	$ϕ_{h} (t)$	1
Deactivation coefficient	$ϕ_{c} (t)$	1

Identification phase

By fixing one of its two inputs, the CSTR is considered as a single input single output (SISO) system. the coolant flow rate is fixed $q_{c} = 70 lmi n^{- 1}$ and the volumetric flow rate q is chosen as the variable input. The CSTR is characterized by a sampling time $T = 1 \min$ . We choose the concentration $C_{A}$ of the reactant A as output of the CSTR.

The input signal is divided into two intervals as shown in Figure 3:

$[1 : 200]$ : time spent on the identification step.

$[201 : 300]$ : time spent on the validation step.

Figure 3.

Input: feed flow rate.

The GA is used to optimize the five Laguerre poles. A crossover probability of $0.9$ is used. The mutation probability is set to $0.033$ . For selection, the algorithm employs the Stochastic Universal Sampling (SUS) strategy, which ensures a fair and diverse selection process by giving individuals chances proportional to their fitness while maintaining population diversity. For a starting population size of $60$ and a generational count of 5, the optimal Laguerre pole values are as follows: $ξ_{u} = 0.3943$ , $ξ_{y} = 0.0916$ , $ξ_{uu} = 0.8153$ , $ξ_{yy} = 0.0306$ , and $ξ_{yy} = 0.8416$ . The identification of the Fourier coefficients using the Recursive Least Squares (RLS) method yields the following vector: $C = [0.0002 0.0003 0.7549 0.0538 0.0684 0.0346 0.0551$ $- 0.0009 0.0002 0.0006 - 0.0003]$ . The modeling performance is shown in Figure 4 where we illustrate the evolution of the system output and the NARX-Laguerre model output.

Figure 4.

Validation.

Application of the proposed PID-NMPC

This section presents the application of the nonlinear predictive control algorithm, based on the NARX-Laguerre model with reduced parametric complexity, to regulate the concentration in the CSTR benchmark, also represented by the NARX-Laguerre model. The key reasons for the superior performance of our proposed approach are first the limitations of the PID controller (PID controllers are simple but lack predictive capabilities, leading to suboptimal performance in systems with strong nonlinearities), second, weaknesses of the NMPC (despite the NMPC provides optimal control by solving future states, its computational complexity can lead to slower responses in real-time applications), and finally, the PID-NMPC combines fast response of PID (for immediate disturbances) with the predictive optimization. The hybrid structure compensates for model uncertainties by leveraging PID’s robustness while maintaining optimality via NMPC.

The simulation settings are fixed in the following way: $N_{p} = 7$ , $μ_{1} = 0.5$ , $μ_{2} = 0.2$ , and $μ_{3} = 0.2$ . The system’s physical limitations are defined by the following inequality:

\begin{matrix} 70 \leq u (k + j) \leq 85 \end{matrix}

(34)

Note: The Sequential Quadratic Programming (SQP) technique is used to solve our nonlinear optimization problem with constraints. This method is iterative and typically begins by making a quadratic approximation of the optimization problem, then iteratively adjusts this approximation to get closer to the optimal solution.

Figure 5 shows the control signal that complies with the constraints specified in relation (34). In Figure 6, we present both the reference trajectory and the output from the CSTR benchmark. The figure clearly illustrates the agreement between the two outputs, demonstrating the efficacity of the enhanced PID control algorithm proposed in this paper.

Figure 5.

Control signal.

Figure 6.

Validation.

Figure 8 illustrates that the PID NMPC controller achieves better performance than both the PID and NMPC controllers.

Results and discussion

Figure 5 demonstrates that the predicted control signal adheres to the proposed constraints, ensuring the system operates within the desired limits. Figure 6 illustrates that the concentration effectively tracks the setpoint, confirming the robustness of the control strategy. Figure 7 reveals that the optimization criterion exhibits two peaks during the setpoint changes, indicating the system’s dynamic response to transitions. Figure 8 highlights the efficiency of the predictive PID NARX-Laguerre control compared to classical PID and NMPC based on NARX-Laguerre, particularly in terms of response time and tracking quality. In addition, Table 2 summarizes the advantages of the proposed control method over the other two, showcasing superior performance in terms of peak overshoot and settling time. These results collectively validate the effectiveness of the proposed predictive PID NARX-Laguerre control strategy.

Figure 7.

Minimization criterion.

Figure 8.

Controllers.

Table 2.

Transient performance analysis.

Controller	Peak overshoot (%)	Setting Time (minutes)
PID	$17.96$	1
NMPC NARX-Laguerre	0	3
PID based on NARX-Laguerre	$8.9$	$0.4$

Conclusion

This paper introduces a novel approach to predictive PID control for nonlinear systems, using the NARX-Laguerre model, which provides the benefit of lower parametric complexity. The NARX-Laguerre model is initially presented, along with its recursive form. To identify the model, the GA is applied to set the five Laguerre poles in their optimal values, and the RLS method is used for determining the Fourier coefficients. Then, the proposed PID NMPC framework based on this model is developed. The effectiveness of the proposed PID NMPC algorithm is validated through extensive testing on the CSTR benchmark, demonstrating its efficiency in nonlinear control applications. While the results are promising, some limitations should be acknowledged. The current approach assumes a static offline-identified model, which may not be suitable for highly time-varying or noisy systems. In addition, the computational cost of genetic optimization could hinder real-time adaptation in fast processes. Future work could explore adaptive or online identification of the NARX-Laguerre model, as well as more efficient optimization techniques to enhance real-time feasibility.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Imen Ben Abdelwahed

Mohamed Guerfel

Kais Bouzrara

Data availability

Data sharing is not applicable to this article, as no data sets were generated or analyzed during the current study.

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